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First, two simple principles: (A) For any given dimension, treat the next highest dimension as its “time.” (B) Mapping out how the state of a given dimension changes over “time” gives you a visualization of the next highest dimension.

To make it easier, I start from the smallest and work highest. Going the other way hurts my head.

To keep it short, I’ll run through 8 or 9 dimensions. The three dimensions we live in are numbers 4, 5 and 6. (Dimension 6 is the really interesting one to me.)

0. Start with a singularity of zero dimension. It’s just a reference point, such as the intersection of two lines or three planes. It’s a point.

1. Map out how that point changes over time. You now have a 1-dimensional line.

2. Map out how that line changes over time. You now have a 2-dimensional plane.

3. Map out how that plane changes over time. You now have a 3-dimensional space.

4. Map out how that space changes over time. Imagine each new instance of that space as an infinitesimal point. You now have a 4-dimensional line, with each point on that line being how the entire 3-dimensional space was at any given moment.

5. Map out how that 4-dimensional line changes over its time, and you now have a 5-dimensional plane.

6. Map out how that 5-dimensional plane changes over its time, and you now have a 6-dimensional field. Think of this as what we normally think of as the 3-dimensional space we experience.

Each infinitesimal point in that 6-dimensional field is a 3-dimensional space of its own. This helps me visualize how nearby points in a field react as they do -- they’re not different things spookily acting at a distance, but are instead just the same 3-dimensional sphere (or whatever) reacting to something over its “time.”

7. Map out how that 6-dimensional field changes over time, and you now have a 7-dimensional thing that we refer to as “spacetime.”

8. I need another dimension on top of that, an 8th dimension, for a sort of Bohm pilot wave that helps me imagine a photon trying out various pathways in its phase space, outside of time, perhaps interfering with each other as in a 2-slit experiment, then traveling down one of those pathways in the spacetime we experience. (See discussion at https://www.physicsforums.com/showthread.php?t=148050&page=5 )

Like I said, I’m not in any way saying this is RIGHT, just that it helps me deal with the concepts. I’d love to hear what works for others.